from Robert Wermuth, author of "Wermuth's GREEKBOOK"

When you see a Greek word that looks like a verb at its beginning that has declined endings, that word must be a Participle. It is critically important that the beginning Greek student learn the declension of the 3rd Declension noun ἀρχῶν, ὁ (ruler). From this noun active participle endings are derived, and the noun’s stem (ἀρχ-) is replaced with a verbal stem (or, as I like to say in class, “We’re putting Noah back into the ‘ark’ [ἀρχ-] and sending him on a vacation.” The resultant verb stem + declined endings = a verbal-adjective, the grammatical description for a participle (e.g., λύων, λύοντος…).

When studying and memorizing participle endings, it is helpful to remember the nominative singular forms of each of the active, middle and passive endings that occur. Including thematic vowel changes, there are four active forms of participle endings (one of them used “passively” with Aorist passives) and three middle/passive forms. Since participles are verbal adjectives, the student will observe a verbal stem with declined endings in all three genders. The active endings (nominative singular) are represented by: -ων, -ουσα, -ον (Present, 2nd Aorist); -ας, -ασα, -αν (1st Aorist); -εις, -εισα, -εν (1st & 2ndAorist passive; remember: Aorist Passives always utilize “active looking” endings); and -ως, -υια, -ος (Perfect). Since each of these groups of endings are declined in a similar way with the masculines and neuters in the 3rd declension and the feminines in the 1st declension (like γλῶσσα), knowledge of the nominative singulars becomes a “jumping off” point in the student’s mind for recognition of any other case forms he may encounter. The middle/passive endings are much more simple, occuring in the 2nd declension for masculine and neuter forms, and the 1st declension for the feminines (like ἀγάπη) and represented by; -ομενος, –ομενη, –ομενον (Present, 2nd Aorist); -αμενος, -αμενη, -αμενον (1st Aorist); –μενος, -μενη, -μενον (Perfect).

When the Greek “circumstantial” (temporal, adverbial) participle occurs (without an article) in the Present tense. The “temporal” (time-oriented) adverb used to translate a Present tense circumstantial participle is “As” (or “While”). The time frame of the “dependent” (temporal, circumstantial) clause should be consistent with the time frame of the indicative verb of the “independent” (main) clause.

When the Greek “circumstantial” participle occurs in the Aorist tense. The “temporal” (time-oriented) adverb used to translate an Aorist tense circumstantial participle is “After.” Again, the time frame of the “dependent” (temporal, circumstantial) clause is translated concurrent with the time frame of the indicative verb in the “independent” (main) clause. (ibid, p. 69)

(AORIST) ἐλθῶν οὖν ὁ Ἰησους εὖρεν αὐτὸν.“Therefore after He came Jesus found him.” (John 11:17a)
(Better): “Therefore after Jesus came He [Jesus] found him.”